# Pressure trace of a tank fed by a compressor

• Sweepy
In summary, the problem is to determine the pressurisation rate of a tank being filled by a pipe connected to a compressor. The attempt at a solution uses the equation of state for both the tank and the pipe and takes the time derivative to find the rate of molecular flow into the tank. The relationship between the rate of molecular flow into the tank and the pressure within the tank is found using the equation of state and the linking relationship. Using the relationship between the rate of molecular flow into the tank and the pressure within the tank, the pressure of the tank can be determined.
Sweepy
G'Day All,

This is my first post so please let me know if I have completed this form incorrectly, or missed a point of etiquette etc...

1. Homework Statement

The problem is to determine the pressurisation rate of a tank being filled by a pipe connected to a compressor.

Assumptions:
• Pipe friction is negligible
• Pipe volumetric flow rate is known
• Tank volume is constant
• Air is the fluid moving through the system and is treated as an ideal gas
• The temperature of the fluid is constant

## Homework Equations

Equation of State:
$$pV=nRT$$

$$\dot{n}_{tank} = \dot{n}_{pipe}$$

The rate of molecular flow into the tank is that coming from the pipe

## The Attempt at a Solution

Using the equation of state for both the tank and the pipe and taking the time derivative we have the following.

Tank:
$$\frac{d(pV)}{dt}=\frac{d(nRT)}{dt}$$
$$\dot{p}V=\dot{n}RT$$ as V and T are constant

Pipe:
$$\frac{d(pV)}{dt}=\frac{d(nRT)}{dt}$$
$$\dot{n} = \frac{p\dot{V}}{RT}$$
NB: I've assumed that the pressure is time-invariant which I am not sure is valid. Will return to this

Therefore, combining the two equations yields the following relationship for the system.

$$\dot{p}_{tank} = \frac{p_{pipe}\dot{V}_{pipe}RT_{tank}}{V_{tank}RT_{pipe}}$$
$$\dot{p}_{tank} = \frac{p_{pipe}\dot{V}_{pipe}}{V_{tank}}$$

Here is where I get stuck. If the volumetric flow rate (##\dot{V}##) is inversely proportional to the pressure within the tank (i.e. ##\dot{V}\propto\frac{1}{p_{tank}}##; this is what I believe based on compressor performance curves), then the relationship can be re-written as

$$\dot{p}_{tank} = \frac{p_{pipe}}{V_{tank}}\frac{C}{p_{tank}}$$
Where C is a constant

So,
$$\dot{p}_{tank} = \frac{\alpha}{p_{tank}}$$
Where ##\alpha=\frac{p_{pipe}C}{V_{tank}}##

This is an seperable 1st order ODE which we can re-write as

$$\frac{dp}{dt} = \frac{\alpha}{p}$$
$$p\cdot dp = \alpha\cdot dt$$
$$\int p\cdot dp = \int \alpha\cdot dt$$
$$\frac{p^{2}}{2} = \alpha t + C$$, where C is a constant
$$p = \pm \sqrt{2\alpha t + 2C}$$, we reject the '-' value as negative pressure is unphysical

Finally, we have

$$p = \sqrt{2\alpha t + 2C}$$

This plots a pressure-time trace that I would expect, that is, one which flattens over time as the pressure in the tank increases creating a resistance (back pressure) to the compressor. But this is based on the assumption of the volumetric flow rate being inversely proportional to the pressure in the tank which I do not know for sure.

Additionally, I am not sure about my earlier assumption that the pressure exiting the pipe is constant. My gut tells me that this should be equal to the pressure of the tank (excluding any wacky compressibility effects) as they are connected directly

If that were the case, and the pipe pressure was equivalent to the tank pressure, then the inverse proportionality of the volumetric flow rate would cancel with this new pipe pressure (both being equal to ##p_{tank}##) and would result in a linear pressure trace.

This doesn't feel right to me intuitively, but I thought I would put it up here to get some outside comment on my thinking/methodology

Hello Sweepy,

I see some internal inconsistencies popping up:
Sweepy said:
Using the equation of state for both the tank and the pipe
for the pipe the volumetric flow rate is known. Is it constant ?

Sweepy said:
I've assumed that the pressure is time-invariant which I am not sure is valid.
What pressure ? Surely not the pressure for which you want to find the time dependence ?

For the pipe:
If you have ##pV = nRT## and ##\dot V## is constant and given, ##T## is constant, you have ##\dot n = {p \dot V\over RT}\ \ ##.

For the tank:
With the tank volume constant I would expect ##{\displaystyle \dot p = {\dot n RT \over V }= {p\dot V\over V}}\ \ ##.

If you have no pump curve, that's the best you can do over a limited period of time.

## 1. What is a pressure trace of a tank fed by a compressor?

A pressure trace is a graphical representation of the changes in pressure inside a tank over a period of time. It is typically measured in units of pressure, such as pounds per square inch (psi) or kilopascals (kPa). In this case, the tank is being fed by a compressor, meaning that the pressure is being increased by the compressor to fill the tank.

## 2. How is a pressure trace of a tank fed by a compressor useful?

A pressure trace can provide valuable information about the functioning of the tank and compressor system. It can help identify any irregularities or fluctuations in pressure, which could indicate potential issues with the equipment. It can also be used to monitor and optimize the performance of the system.

## 3. What factors can affect the pressure trace of a tank fed by a compressor?

Several factors can impact the pressure trace of a tank fed by a compressor. These include the size and type of tank, the capacity and efficiency of the compressor, the temperature and humidity of the environment, and any potential leaks or malfunctions in the system.

## 4. How can the pressure trace of a tank fed by a compressor be measured?

The pressure trace can be measured using a pressure gauge or transducer, which converts the pressure into an electrical signal that can be recorded and displayed graphically. The gauge or transducer is typically connected to the tank and calibrated to accurately measure the pressure inside.

## 5. What can be done to maintain a stable pressure trace in a tank fed by a compressor?

To maintain a stable pressure trace, it is important to regularly check and maintain the equipment. This may include checking for leaks, ensuring proper lubrication and maintenance of the compressor, and monitoring the pressure and temperature levels inside the tank. It is also important to follow proper safety protocols and guidelines when working with a tank and compressor system.

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